Productivity Index for Darcy and pre-/post-Darcy Flow (Analytical Approach)

TL;DR

This paper develops an analytical framework to evaluate the well productivity index considering nonlinear flow regimes (pre-Darcy, Darcy, post-Darcy) in reservoirs, highlighting the effects of low and high velocity flows on well productivity.

Contribution

It introduces a unified mathematical model combining all three flow regimes to analytically assess the impact on productivity index, including the often overlooked pre-Darcy effects.

Findings

Pre-Darcy effects significantly influence the productivity index at low velocities.

The analytical formulas account for critical transitional velocities between flow regimes.

Flow regime transitions affect well productivity depending on flux and flow parameters.

Abstract

We investigate the impact of nonlinearity of high and low velocity flows on the well productivity index (PI). Experimental data shows the departure from the linear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells and fractures is described by Forchheimer equations and is relatively well-studied. While low velocity flow receives much less attention, there is multiple evidence suggesting the existence of pre-Darcy effects for slow flows far away from the well. This flow is modeled via pre-Darcy equation. We combine all three flow regimes, pre-Darcy, Darcy and post-Darcy, under one mathematical formulation dependent on the critical transitional velocities. This allows to use our previously developed framework to obtain the analytical formulas for the PI for the cylindrical reservoir. We study the impact of pre-Darcy effect on the PI of steady-state flow depending on the well-flux and the parameters of the equations.